If the ergodic transformations S, T generate a free ${\mathbb{Z}}^{2}$ action on a finite non-atomic measure space (X,S,µ) then for any ${c}_{1},{c}_{2}\in \mathbb{R}$ there exists a measurable function f on X for which ${\left(N+1\right)}^{-1}{\sum}_{j=0}^{N}f\left({S}^{j}x\right)\to {c}_{1}$ and ${(N+1)}^{-1}{\sum}_{j=0}^{N}f\left({T}^{j}x\right)\to {c}_{2}\mu $-almost everywhere as N → ∞. In the special case when S, T are rationally independent rotations of the circle this result answers a question of M. Laczkovich.

In this paper we give a complete answer to the famous gradient problem of C. E. Weil. On an open set G ⊂ R we construct a differentiable function f: G → R for which there exists an open set Ω ⊂ R such that ∇f(p) ∈ Ω for a p ∈ G but ∇f(q) ∉ Ω for almost every q ∈ G. This shows that the Denjoy-Clarkson property does not hold in higher dimensions.

Motivated by the concept of tangent measures and by H. Fürstenberg’s definition of microsets of a compact set $A$ we introduce micro tangent sets and central micro tangent sets of continuous functions. It turns out that the typical continuous function has a rich (universal) micro tangent set structure at many points. The Brownian motion, on the other hand, with probability one does not have graph like, or central graph like micro tangent sets at all. Finally we show that at almost all points Takagi’s...

It is known that for almost every (with respect to Lebesgue measure) a ∈ [√2,2] the forward trajectory of the turning point of the tent map ${T}_{a}$ with slope a is dense in the interval of transitivity of ${T}_{a}$. We prove that the complement of this set of parameters of full measure is σ-porous.

Suppose $F\subset [0,1]$ is closed. Is it true that the typical (in the sense of Baire category) function in ${C}^{1}[0,1]$ is one-to-one on $F$? If ${\underline{dim}}_{B}F<1/2$ we show that the answer to this question is yes, though we construct an $F$ with ${dim}_{B}F=1/2$ for which the answer is no. If ${C}_{\alpha}$ is the middle-$\alpha $ Cantor set we prove that the answer is yes if and only if $dim\left({C}_{\alpha}\right)\le 1/2.$ There are $F$’s with Hausdorff dimension one for which the answer is still yes. Some other related results are also presented.

We study the relationship between derivates and variational measures of additive functions defined on families of figures or bounded sets of finite perimeter. Our results, valid in all dimensions, include a generalization of Ward’s theorem, a necessary and sufficient condition for derivability, and full descriptive definitions of certain conditionally convergent integrals.

Let $fH\subset \mathbb{R}\to \mathbb{R}$ be $k$ times differentiable in both the usual (iterative) and Peano senses. We investigate when the usual derivatives and the corresponding Peano derivatives are different and the nature of the set where they are different.

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